Differential equations and geometric structures on manifolds

概括地说，我的研究建议是关于曲面空间上的微分方程和几何结构之间的关系。该提案有五个部分，它们探索了这一主题的具体方面，部分是通过由数学问题和未来物理的猜想引发的问题，部分是通过更基本的几何性质的问题。在这些问题上的进展应该有助于提高我们对曲面空间上微分方程解的性态的理解，有助于进一步阐明一些现有的猜想，也有助于开辟研究的新途径。我们现在列出我们提案的五个组成部分。第一个项目研究了高维轴对称黑洞几何中波的长期动力学，建立在某种同齐性的Sasaki-Einstein度规上的ADS/CFT对应，以及Kerr几何中的纠缠。第二部分是关于微分算子的谱问题可以用代数方法至少部分地解决的例子，以及与正交多项式系统和可积系统的联系。第三部分是关于流形上的双环几何的构造，部分灵感来自于它们在广义相对论中的洛伦兹类似。四是利用Leray关于Cauchy-Kovalevskaa型系统的分枝Cauchy问题的理论，推广了对合解析外微分系统积分流形的Cartan-Kaehler存在定理。最后，第五个项目是利用Adler和Taylor首创的游程集几何技巧，探索三维或更高维空间上随机度量的Ricci曲率的行为。

Stated in general terms, my research proposal is concerned with the relationship between differential equations and geometric structures on curved spaces. The proposal has five components which explore specific aspects of this theme, partly through questions motivated by mathematical problems and conjectures coming Physics and partly through problems which are of a more fundamentally geometric nature. Progress on these questions should help improve our understanding of the behavior of solutions of differential equations on curved spaces, help shed further light on some existing conjectures, and also help open new avenues for research. We now list the five components of our proposal. The first project deals with the long term dynamics of waves in higher dimensional axisymmetric black hole geometries, with the AdS/CFT correspondence for the geometries built on certain cohomogeneity one Sasaki-Einstein metrics and with entanglement in Kerr geometry. The second is concerned with instances in which spectral problems for differential operators can be solved at least partially by algebraic methods, and the connections with orthogonal polynomial systems as well as integrable systems. The third deals with the construction of ambi-toric geometries on manifolds and orbifolds, inspired in part by their Lorentzian analogues in General Relativity. The fourth is an attempt to extend the Cartan-Kaehler existence theorem for the existence of integral manifolds of involutive analytic exterior differential systems by using Leray's theory of the ramified Cauchy problem for systems of Cauchy-Kovalevskaia type. Finally the fifth project is an exploration of the behavior of the Ricci curvature for random metrics on spaces of dimension three or higher, using the techniques on the geometry of excursion sets pioneered by Adler and Taylor.